Assessing the validity of the thermodynamic uncertainty relation in quantum systems

We examine the so-called thermodynamic uncertainty relation (TUR), a cost-precision trade-off relationship in transport systems. Based on the fluctuation symmetry, we derive a condition for invalidating the TUR for general nonequilibrium (classical and quantum) systems. We find that the first nonzero contribution to the TUR beyond equilibrium, given in terms of nonlinear transport coefficients, can be positive or negative, thus affirming or violating the TUR depending on the details of the system. We exemplify our results for noninteracting quantum systems by expressing the thermodynamic uncertainty relation in the language of the transmission function. We demonstrate that quantum coherent systems that do not follow a population Markovian master equation, e.g., by supporting high-order tunneling processes or relying on coherences, violate the TUR.

Figure 1

Transport models that show violations of the TUR in different regimes: model A, a single-dot junction; model B, a serial double-dot junction; and model C, a side-coupled double-dot junction.

Figure 2

Transmission functions for models A, B, and C at (a) weak and (b) strong couplings to the metal leads. Model A, single-dot junction: ${\epsilon }_d=0$. Model B, serial double-dot junction: ${\epsilon }_{L,R}=0$ and $\mathrm{\Omega }=0.1$. Model C, side-coupled double-dot junction: ${\epsilon }_{1,2}=0$ and $\mathrm{\Omega }=0.1$.

Figure 3

Single quantum dot: Violation of the TUR ($\mathcal{F}<0$), illustrated by varying the (a) metal-molecule coupling $\mathrm{\Gamma }$ and (b) voltage. Parameters are ${\epsilon }_d=0$, $\beta =1$, and $\mathrm{\Gamma }=4$ in panel (b). Exact calculations for $\mathcal{F}/V^2$ are based on Eq. (9). These are compared to the analytic formula (13); we display ${\mathrm{C}}_{\mathrm{neq}}/G_1$ in panel (a) and $V^2{\mathrm{C}}_{\mathrm{neq}}/G_1$ in panel (b).

Figure 4

Serial double quantum dot: (a) Violation of the TUR ($\mathcal{F}<0$) within a certain range for $\mathrm{\Gamma }$. The transmission function is given in Eq. (26), with ${\epsilon }_L={\epsilon }_R=0$, $\beta =1$, $\mathrm{\Omega }=0.05$, and $\mathrm{\Gamma }=0.08$. Exact calculations for $\mathcal{F}/V^2$, based on Eq. (9), are compared to results from the analytic formula for ${\mathrm{C}}_{\mathrm{neq}}/G_1$ based on Eq. (13). (b) When the ratio ${\mathcal{T}}_2/{\mathcal{T}}_1$ (solid line) exceeds 2/3 (dashed line), the TUR is violated at weak coupling as shown in panel (a).

Figure 5

Serial double quantum dot: (a) Violation of the TUR ($\mathcal{F}<0$) as a function of bias. The transmission function, given in Eq. (26), is evaluated at ${\epsilon }_L={\epsilon }_R=0$, $\beta =1$, and $\mathrm{\Omega }=0.05$. (b) The exact charge current $\langle j\rangle$, the associated noise $\langle \langle j^2\rangle \rangle$, and the TUR function $\mathcal{F}/V^2$ plotted against $\mathrm{\Gamma }$. Here $V=5$. Other parameters are the same as those in panel (a).

Figure 6

Uncovering TUR violations in Model B ($\mathrm{\Omega }/\mathrm{\Gamma }\sim 1$) and Model C ($\mathrm{\Omega }/\mathrm{\Gamma }\ll 1)$. We use $\beta =1$, $V=1$, and $\mathrm{\Gamma }=4$.